Let's break this down categorically:
with respect to finances
A capital-D Democrat in the House of Representatives can generally expect to receive money from the Democratic Congressional Campaign Committee. A Democratic senator can analogously expect to receive money from the Democratic Senatorial Campaign Committee. Both can expect to receive additional funds from various other sources within and without the Democratic party, including individual voters, state and local party committees, personal contributions by the candidate, and "non-super" political action committees ("super PACs" can raise and spend unlimited amounts of money on ads, but cannot coordinate directly with candidates or give them money). This entire apparatus is mirrored on the Republican side, and by most third parties (albeit not always at all levels, e.g. a small party may not have a national committee, or may not have local committees in every state or municipality).
A true independent can expect to receive funds from some of these sources, possibly including the party apparatus if one of the parties likes them enough (though typically not if that party already has another candidate in that race - see for example Sen. Sanders, who twice declined Democratic nominations after winning the Democratic Senate primary in Vermont, precluding another Democrat from running). Depending on their politics and viability (likelihood of winning), they may or may not receive help from PACs. Many independents have run primarily on some combination of individual donations and the candidate's personal wealth.
This table summarizes the amounts of money that different groups can legally spend on political campaigns. The committees generally try to "triage" their money, first focusing on winnable but at-risk campaigns. Senators and representatives in "safe" seats can still expect a decent amount of fundraising, but they will not be the top priority of the national committees.
voting discipline
A Democratic senator will be a member of the Democratic Caucus. Currently, two independents (Bernie Sanders and Angus King) are also members of this caucus. Caucus members are generally expected to vote as a bloc on significant legislation (as designated by party leadership). The strength of this expectation has ebbed and flowed considerably over the history of American politics. It is currently very strong, because both Houses of Congress are heavily polarized. Senators or representatives can also form smaller caucuses to advance more specific political goals than those pursued by the party caucuses. When these come into conflict with the larger party caucuses, it can greatly impede the ability of party leadership to accomplish its objectives.
The whip is responsible for enforcing party discipline. Because the United States does not use the Westminster system, this cannot be done with cabinet positions (which are under the control of the president, and can't legally be members of Congress anyway) and instead normally revolves around congressional committee membership and chairmanship. In extreme cases, a senator or representative who refuses to vote the party line could be expelled from the caucus altogether. This is quite rare, but would obviously affect their eligibility for the fundraising efforts discussed above.
As I mentioned, independents can join the caucuses as well as party members. This is allowed because the caucus benefits by gaining the independent's vote on important issues, and the independent in turn gets to participate in the caucus's decision making process. Should the caucus and the independent develop widely divergent political beliefs, they might go their separate ways, but this is unlikely unless the independent wants to join the other party's caucus. If an independent is participating in neither party caucus, that will greatly weaken their ability to influence legislation.
allocation of your time etc.
This is the most slippery item. In general, they are at a minimum expected to show up for votes which the party leadership cares about, to caucus meetings, and to various informal and semi-formal events (often, but not always, connected to fundraising). There is also an expectation that they will be less available during campaign season (autumn of even-numbered years, for all representatives and one-third of the Senate at a time), during which time Congress usually recesses. Congress takes several other recesses throughout the year, with similar expectations.
However, if a legislator were to be persistently and obnoxiously unavailable for informal consultations with their fellow party members, it would likely cause problems. They need to communicate freely with one another to coordinate votes and avoid wasting time on doomed measures. For example, the failure of the Republicans' "skinny repeal" act was all the more surprising because it happened in an official, on-the-record vote. Normally, it would have failed behind closed doors (Sens. McCain, Collins, and Murkowski would have told Majority Leader McConnell that he did not have their votes, and there it would have ended).
